No, Henry VIII is at Windsor, John is in somewhere like Peterborough, (he died at Newark). Regardless of if you think he was a good or bad king /person, IF the bones can be reasonably identified as his then yes, he should have a royal tomb, though ideally at minimal public expense - maybe the Richard III society could raise funds?

I'm confused about this, aren't the bones supposed to show signs of a hunchback when in reality he didn't have one? It was only Shakespeare who gave him the hunchback, he was apparently considered to be a very handsome man and very strong. He was married at York, crowned Prince of Wales there and grew up at Middleham Castle so if it is him York would be the choice.

Lots of historians argue Tudor propagandists were responsible for Richard's "evil" appearance. You can track the changes in his physical appearance in paintings etc as the propaganda against him becomes more extreme. Shakespeare was in the pay of a Tudor monarch. (vague memories of a course at university many years ago...)